17,940 research outputs found
Tischler graphs of critically fixed rational maps and their applications
A rational map on the Riemann
sphere is called critically fixed if each critical point
of is fixed under . In this article we study properties of a
combinatorial invariant, called Tischler graph, associated with such a map.
More precisely, we show that the Tischler graph of a critically fixed rational
map is always connected, establishing a conjecture made by Kevin Pilgrim. We
also discuss the relevance of this result for classical open problems in
holomorphic dynamics, such as combinatorial classification problem and global
curve attractor problem
Non-integrability of measure preserving maps via Lie symmetries
We consider the problem of characterizing, for certain natural number ,
the local -non-integrability near elliptic fixed points of
smooth planar measure preserving maps. Our criterion relates this
non-integrability with the existence of some Lie Symmetries associated to the
maps, together with the study of the finiteness of its periodic points. One of
the steps in the proof uses the regularity of the period function on the whole
period annulus for non-degenerate centers, question that we believe that is
interesting by itself. The obtained criterion can be applied to prove the local
non-integrability of the Cohen map and of several rational maps coming from
second order difference equations.Comment: 25 page
Dessins d'enfants and Hubbard Trees
We show that the absolute Galois group acts faithfully on the set of Hubbard
trees. Hubbard trees are finite planar trees, equipped with self-maps, which
classify postcritically finite polynomials as holomorphic dynamical systems on
the complex plane. We establish an explicit relationship between certain
Hubbard trees and the trees known as ``dessins d'enfant'' introduced by
Grothendieck.Comment: 27 pages, 8 PostScript figure
Hamiltonian mappings and circle packing phase spaces
We introduce three area preserving maps with phase space structures which
resemble circle packings. Each mapping is derived from a kicked Hamiltonian
system with one of three different phase space geometries (planar, hyperbolic
or spherical) and exhibits an infinite number of coexisting stable periodic
orbits which appear to `pack' the phase space with circular resonances.Comment: 23 pages including 12 figures, REVTEX
Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems
This is an updated and expanded version of our earlier survey article
\cite{Gut5}. Section introduces the subject matter. Sections expose the basic material following the paradigm of elliptic, hyperbolic and
parabolic billiard dynamics. In section we report on the recent work
pertaining to the problems and conjectures exposed in the survey \cite{Gut5}.
Besides, in section we formulate a few additional problems and
conjectures. The bibliography has been updated and considerably expanded
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